THE INCLUSION RELATIONS OF THE COUNTABLE MODELS OF SET THEORY ARE ALL ISOMORPHIC JOEL DAVID HAMKINS AND MAKOTO KIKUCHI M Abstract. The structures M, ⊆ arising as the inclusion rela- M tion of a countable model of sufficient set theory M, ∈ , whether M well-founded or not, are all isomorphic. These structures M, ⊆ are exactly the countable saturated models of the theory of set- theoretic mereology: an unbounded atomic relatively complemented distributive lattice. A very weak set theory suffices, even finite set theory, provided that one excludes the ω-standard models with no infinite sets and the ω-standard models of set theory with an amor- phous set. Analogous results hold also for class theories such as G¨odel-Bernays set theory and Kelley-Morse set theory. 1. Introduction Set-theoretic mereology is the study of the inclusion relation ⊆ as it arises within set theory. In any set-theoretic context, with the set membership relation ∈, one may define the corresponding inclusion relation ⊆ and investigate its properties. Thus, every model of set theory M, ∈M gives rise to a corresponding model of set-theoretic mereology M, ⊆M , the reduct to the inclusion relation. In our previous article [HK16], we identified exactly the complete M M theory of these mereological structures M, ⊆ . Namely, if M, ∈ is a model of set theory, even for extremely weak theories, including set theory without the infinity axiom, then the corresponding mereological M arXiv:1704.04480v1 [math.LO] 14 Apr 2017 reduct M, ⊆ is an unbounded atomic relatively complemented dis- tributive lattice. We call this the theory of set-theoretic mereology. By a quantifier-elimination argument that we give in [HK16], partaking of Tarski’s Boolean-algebra invariants and Erˇsov’s work on lattices, this theory is complete. This work was supported by JSPS KAKENHI Grant Number 17H02263. The research project builds on our earlier paper [HK16]. This segment of the work began in Kyoto at the conference Mathematical Logic and its Applications, organized by the second author and held at the Research Institute for Mathematical Sciences at Kyoto University in September 2016. The first author is grateful for the support of his participation there. Commentary concerning this paper can be made at jdh.hamkins.org/inclusion-relations-are-all-isomorphic. 1 2 HAMKINSANDKIKUCHI After that work, we found it natural to inquire: Question 1. Which models of set-theoretic mereology arise as the in- clusion relation ⊆ of a model of set theory? More precisely, given a model hM, ⊑i of set-theoretic mereology, un- der what circumstances can we place a binary relation ∈M on M in M such a way that M, ∈ is a model of set theory and the inclusion M relation ⊆ defined in M, ∈ is precisely the given relation ⊑? One can view this question as seeking a kind of Stone-style representation of the mereological structure hM, ⊑i, because such a model M would provide a representation of hM, ⊑i as a relative field of sets via the model of set theory M, ∈M . A second natural question was to wonder how much of the theory of the original model of set theory can be recovered from the mereological reduct. M Question 2. If M, ⊆ is the model of set-theoretic mereology arising M as the inclusion relation ⊆ of a model of set theory M, ∈ , what part of the theory of M, ∈M is determined by the structure M, ⊆M ? In the case of the countable models of ZFC, these questions are completely answered by our main theorems. Main Theorems. (1) All countable models of set theory M, ∈M |= ZFC have iso- morphic reducts M, ⊆M to the inclusion relation. (2) The same holds for models of considerably weaker theories such as KP and even finite set theory ZF¬∞, provided one excludes the ω-standard models without infinite sets and the ω-standard models having an amorphous set. (3) These inclusion reducts M, ⊆M are precisely the countable saturated models of set-theoretic mereology. (4) Similar results hold for class theory: all countable models of G¨odel-Bernays set theory have isomorphic reducts to the inclu- sion relation, and this reduct is precisely the countably infinite saturated atomic Boolean algebra. Specifically, in theorem 5 we show that the mereological reducts M, ⊆M of the models of sufficient set theory are always ω-saturated, and from this it follows on general model-theoretic grounds (corollary 6) that they are all isomorphic, establishing statements (1) and (2). So a countable model hM, ⊑i of set-theoretic mereology arises as the in- clusion relation of a model of sufficient set theory if and only if it is 3 ω-saturated (corollary 7), establishing (3) and answering question 1. Consequently, in addition, the mereological reducts M, ⊆M of the countable models of sufficient set theory know essentially nothing of the M M theory of the structure M, ∈ from which they arose, since M, ⊆ arises equally as the inclusion relation of other models hM, ∈∗i with any desired sufficient alternative set theory (corollary 8), a fact which an- swers question 2. Our analysis works with very weak set theories, even finite set theory ZF¬∞, provided one excludes the ω-standard models with no infinite sets and the ω-standard models with an amorphous set, since the inclusion reducts of these models are not ω-saturated. In section 4 we prove that most of these results do not generalize to un- countable models, nor even to the ω1-like models, although theorem 12 shows that every saturated model of set-theoretic mereology is realized as the inclusion relation of a model of any desired consistent set theory. Our results have some affinity with the classical results in models of arithmetic concerned with the additive reducts of models of PA. Restricting a model of set theory to the inclusion relation ⊆ is, af- ter all, something like restricting a model of arithmetic to its additive part. Lipshitz and Nadel [LN78] proved that a countable model of Presburger arithmetic (with + only) can be expanded to a model of PA if and only if it is computably saturated. We had hoped at first to prove a corresponding result for the mereological reducts of the models of set theory. In arithmetic, the additive reducts are not all isomor- phic, since the standard system of the PA model is fully captured by the additive reduct. Our main result for the countable models of set theory, however, turned out to be stronger than we had expected, since the inclusion reducts are not merely computably saturated, but fully ω- saturated, and this is why they are all isomorphic. Meanwhile, Lipshitz and Nadel point out that their result does not generalize to uncount- able models of arithmetic, and similarly ours also does not generalize to uncountable models of set theory (see section 4). Another instance of the general phenomenon is known for real-closed fields, since results in [DKS10, DKS12] show that a countable real closed field has an in- teger part that is a model of PA just in case it is either Archimedean or computably saturated. 4 HAMKINSANDKIKUCHI 2. Expressive power of types in set-theoretic mereology Let us begin our analysis by observing that every model of set- theoretic mereology hM, ⊑i can be represented as a relative field of sets, that is, a collection of sets closed under intersection, union and relative complement. This can be seen simply by identifying every ob- ject in M with the set of atoms below it, since one may readily verify that this representation respects the lattice structure of hM, ⊑i. There- fore, allow us freely to use a set-theoretic terminology and notation in set-theoretic mereology, referring to the lattice operations as union, intersection and relative complement. We shall now clarify the exact expressive power of types in set- theoretic mereology. Lemma 3. If p(a1,...,an) is a complete n-type in the language of set-theoretic mereology, then p(a1,...,an) is equivalent over the theory of set-theoretic mereology to the assertions stating for each cell in the Venn diagram of the variables that it has some specific finite size or that it is infinite. a b |a − (b ∪ c)| = ∞ 3 |(a ∩ b) − c| =3 ∞ 2 |b − (a ∪ c)| =2 5 |(a ∩ c) − b| =0 0 ∞ |a ∩ b ∩ c| =5 (∞) |(b ∩ c) − a| = ∞ 17 |c − (a ∪ b)| = 17 c Proof. This is a consequence of the elimination of quantifiers argument from our previous paper [HK16, theorem 9]. We proved that every assertion in the language of set-theoretic mereology is equivalent to a quantifier-free assertion in the language allowing the operations of union ∪, intersection ∩ and relative complement x−y and the relations |τ| = n, which assert that there are precisely n atoms below τ. It follows that a complete type p(a1,...,an) must make such an assertion about every cell in the corresponding Venn diagram of those variables, and furthermore this information determines everything else that one can express about those variables in this language. (Note that in set- theoretic mereology, it follows from unboundedness that the exterior region, which is not represented by any term, must always be infinite.) Although the lemma shows that every type amounts in a sense to finitely many assertions about the cells in the Venn diagram, we are 5 not claiming that every complete type is principal, because the asser- tion that a particular cell in the Venn diagram is infinite, as with the assertion |a−(b∪c)| = ∞ in the diagram above, is not expressible by a single formula in the language of set-theoretic mereology, but rather is expressible in the type as infinitely many assertions stating that that term has no particular finite size.